Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. Plural form of subalgebra.
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Examples

In the 1930s Garrett Birkhoff established the fundamental results of equational logic, namely (1) equational classes of algebras are precisely the classes closed under homomorphisms, subalgebras and direct products, and (2) equational logic is based on five rules: reflexivity, symmetry, transitivity, replacement, and substitution.

Full quantization maps generally only exist on small subalgebras of the full Poisson algebra, and if they do, they need not be unique.

“Maximal beable subalgebras of quantum mechanical observables,”

A central result in this area is the theorem that a lattice arises as the lattice of subalgebras of some algebra if and only if it arises as the lattice of congruences on some algebra.

That is, to construct all the models of the theory of C it suffices to close C first under direct products, then under subalgebras, and finally under homomorphic images; that is, later closures do not compromise earlier ones provided P, S, and H are performed in that order.

Given a class C of algebras, we write P (C) for the class of all algebras formed as direct products of families of algebras of C, S (C) for the class of all subalgebras of algebras of C, and H (C) for the class of all homomorphic images of algebras of C.

Aloc of all local observables into subalgebras which contains physical information about the observables, i.e., it is the net structure of algebras which matters.

Lalgebras closed under subalgebras and direct products.

Lalgebras that includes K and is closed under subalgebras, direct products and homomorphic images.

A class of Lalgebras is a quasivariety if and only if it is closed under subalgebras, direct products and ultraproducts.
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